Hi, The mathematical theory of categories may provide a rational framework for characterizing and analyzing the possible connections between physics, biology, and semiotics. It is for this reason that I begin this post with some definitions in category theory:
(1) “Comparison and analogy are fundamental aspects of knowledge acquisition. We argue that one of the reasons for the usefulness and importance of Category Theory is that it gives an abstract mathematical setting for analogy and comparison, allowing an analysis of the process of abstracting and relating new concepts. This setting is one of the most important routes for the application of Mathematics to scientific problems. “ (Brown and Porter, 2006). (2) “We view a category as giving a fairly general abstract context for comparison. The objects of study are the objects of the category. Two objects, A and B, can be compared if the set C(A,B) is non-empty and various arrows A ---> B are ‘ways of comparing them’. The composition corresponds to: If we can compare A with B and B with C, we should be able to compare A with C.” (Brown and Porter, 2006). (3) “. . . a functor . . is a way of comparing categories, . . . “ (Brown and Porter, 2006). (4) "“Different branches of mathematics (human knowledge; my addition) can be formalized into categories. These categories can then be connected together by functors. And the sense in which these functors provide powerful communication of ideas is that facts and theorems (regularities; my addition) proven in one category (discipline; my addition) can be transferred through a connecting functor to yield proofs of analogous theorems in another category. A *functor** is like a conductor of mathematical truth (*my emphasis*)*” (Spivak, 2013). (A) The scientific controversies surrounding Shannon's entropy H may not be successfully analyzed without taking into account their semiotic aspects. Many would agree that, had von Neumann not suggested to Shannon to name his H function "entropy", a lot of the confusions in the H-S (or information-entropy) debates might have been avoided. (B) There is no doubt that a close formal similarity exists between the mathematical equations of H and S (see Rows 1 and 3, Column 1). But this is a shallow and superficial reason for giving both functions the same name, 'entropy', without first checking that both mathematical functions share some common principles or mechanisms. Since the meaning of 'entropy' in thermodynamics is relatively well established (e.g., a measure of disorder, obeying the Second Law), giving this same name to the H function may lead to unwittingly attributing the same thermodynamic meaning of entropy to H. In fact many prominent scientists and mathematicians unfortunately have taken this road, thereby creating confusions among scholars. (C) A somewhat similar events have transpired during the past 7 years in my research. I, with the help of one of my pre-med students at Rutgers, derived a new equation, called the Planckian distribution equation (PDE) (see Row 3, Column 2), by replacing the universal constants and temperature in the Planck blackbody radiation equation (PBRE) (see Row 1, Column 2) with free parameters, a, b, A and B. For convenience, we define "Planckian processes" as those physicochemical, biomedical or socioeconomic processes that generate numerical data that fit PDE, and there are many such processes found in natural and human sciences [1]. In certain sense, H function of Shannon is related to the S function of Boltzmann as PDE is related to PBRE. Therefore, if there are functors connecting PDE and PBRE (e.g., energy quantization, wave-particle duality) as I strongly believe, it is likely that there can be at least one functor connecting H and S which I do not believe is the Second Law as some physicists and mathematicians claim. As Jon and Stan recently hinted, the functor connecting H and S may well turn out to be "variety" or "complexity" as suggested by Wicken [2, p. 186]. (D) In addition to the "mathematical functors" described in (C), there may be "non-mathematical" or "qualitative" functors connecting H and S on the one hand and PDE and PBRE on the other, and I humbly suggest that these "qualitative functors" may be identified with the Peircean sign triad or semiosis as briefly indicated in Row 6 of the table below. (E) If the contents of Table 1 turn out to be true in principle, we may be justified to recognize two kinds of functors in category theory -- "quantiative" and "qualittive" functors", the former belonging to the domain of mathematics and the latter to that of semiotics. ____________________________________________________________________________________________________ Table 1. A comparison between Shannon's invention of H and my derivation of PDE from Planck radiation equation.-- ____________________________________________________________________________________________________ Shannon's invention of H Ji's derivation of PDE ____________________________________________________________________________________________________ 1. Original Boltzmann equation for entropy S Planck blackbody radiation equation (PBRE) equation (1872-5) (1900) S = k ln W which generalizes as U = ((2 pi h c^2)/lambda^5)/(Exp(hc/kT lambda) - 1) S = - k Sum Pi log Pi ____________________________________________________________________________________________________ 2. Insight or microscopic explanation quantization of action or movement mechanisms of macroscopic measurements (as a prelude to organization) ____________________________________________________________________________________________________ 3. New equation Shannon equation Planckian distribution equation (PDE) invented or invented in 1948 derived in 2008-9 discovered H = - K Sum Pi log_2 Pi y = (a/(Ax + B))^5/(Exp(b/(Ax + B)) - 1) _____________________________________________________________________________________________________ 4. Significance H measures the variety or PDE can be used to measure the degree of complexity of a message non-randomness (also called 'order' or source of a communication 'organization') of a system, called the Planckian system. information (I_P) (2015) ______________________________________________________________________________________________________ 5. Domain of Any field that can generate Any field that can generate long-tailed application probability distributions Pi histograms that fit PDE ______________________________________________________________________________________________________ 6. Semiotic Equation for H was invented but Equation for PDE was derived from PBRE problem H had no 'name'. von Neumann and was found to fit single-molecule enzyme involving "names" recommended for H the same name kinetic data, strongly suggesting that the two and their "objects" 'entropy' used to designate entropy mechanisms of (i) the energy quantification or "referents". S in thermodynamics without and (ii) the particle-wave duality operating atoms knowing whether or not the also operate in enzymes. In other words, the new mechanism or principles underlying equation discovered here has both a rational name S can be extended to H. and reasonable mechanisms suggested by the In other words, the new equation name. Hence the semiotic problem faced by invented by Shannon had no 'genuine' PDE is far less problematic than that faced by H. name nor 'genuine' object or referent. _______________________________________________________________________________________________________ Common principle The Peircean sign triad, or The Peircean sign triad, or or mechanism semiosis semiosis. connecting different disciplines or fields ________________________________________________________________________________________________________ If you have any questions, suggestions, or criticisms, please let me know. All the best. Sung Sungchul Ji, Ph.D. Associate Professor of Pharmacology and Toxicology Department of Pharmacology and Toxicology Ernest Mario School of Pharmacy Rutgers University Piscataway, N.J. 08855 732-445-4701 www.conformon.net References: [1] Ji, s. (2015). Planckian distributions in molecular machines, living cells, and brains: The Wave-particle duality in biomedical sciences. Proceedings of the International Conference on Biology and Biomedical Engineering. Vienna, March 15-17, 2015. Uploaded to ResearchGate in March, 2015. [2] Wicken, J. S. (1987). Entropy and Information: Suggestions for Common Language. Phil. Sci. 54: 176-193.
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